The vector-valued generating-function identity for standard Young tableaux and sorting networks

For $n\geq 2$, let $\operatorname{SYT}(\delta_n)$ be the set of standard Young tableaux of staircase shape $\delta_n$, let $\operatorname{SN}_n$ be the set of sorting networks on $n$ elements, and let $f_t(x_1,\ldots,x_{n-1})$ and $g_s(x_1,\ldots,x_{n-1})$ be the associated generating-function coefficients, with basis vectors $\sigma_t$ and $\pi_s$. The generating-function identity. For $n\geq 2$,

$$\sum_{t\in \operatorname{SYT}(\delta_n)} f_t(x_1,\ldots,x_{n-1})\sigma_t = \sum_{s\in \operatorname{SN}_n} g_s(x_1,\ldots,x_{n-1})\pi_s.$$

This is presented as an algebraic-combinatorial reformulation of the conjectural equality in distribution between the oriented swap process and last passage percolation vectors. Its status is not resolved by the supplied text.